Physics 207 Lecture 12

Transcription

1 Physcs 07 ectue Physcs 07, ectue 6, Oct. 9 Agend: Chpte 3 Cente of ss Toque oent of net ottonl Enegy ottonl oentu Chp. 3: ottonl Dyncs Up untl now otton hs been only n tes of ccul oton wth c v / nd T d v / dt otton s coon n the wold ound us. ny des developed fo tnsltonl oton e tnsfeble. Assgnent: Wednesdy s n ex evew sesson, Ex wll be held n oos B0 & B30 n Vn Vleck t 7:5 P P Hoewok 7, Ch., 5 pobles, NOTE: Due Wednesdy t 4 P P Hoewok 7A, Ch. 3, 5 pobles, vlble soon Physcs 07: ectue 6, Pg Physcs 07: ectue 6, Pg Consevton of ngul oentu hs consequences ottonl Dyncs: A chld s s toy, physcs plygound o student s s nghte A ey-go-ound s spnnng nd we un nd jup on t. Wht does t do? We e stndng on the nd ou fends spn t fste. Wht hppens to us? We e stndng on the wlk towds the cente. Does nythng chnge? How does one descbe otton (gntude nd decton)? Physcs 07: ectue 6, Pg 3 Physcs 07: ectue 6, Pg 4 ottonl Vbles ottonl Vbles... otton bout fxed xs: Consde dsk ottng bout n xs though ts cente:] How do we descbe the oton: θ ecll: At pont dstnce wy fo the xs of otton, the tngentl oton: v x θ v θ α x dθ π (d/s) dt T v Tngentl dx (Anlogous to the lne cse v ) dt α constnt αt 0 (ngul ccelton n d/s (ngul velocty n d/s) θ θ0 0t αt (ngul poston n d) ) α Physcs 07: ectue 6, Pg 5 Physcs 07: ectue 6, Pg 6 Pge

2 Physcs 07 ectue Suy (wth copson to -D D knetcs) Angul α constnt 0 αt θ θ0 0t αt ne constnt v v 0 t x x0 v0t t And fo pont t dstnce fo the otton xs: x θ v α ectue 5, Execse 5 ottonl Defntons A goofy fend sees dsk spnnng nd sys Ooh, look! Thee s wheel wth negtve nd wth ntpllel nd α! Whch of the followng s tue stteent bout the wheel? (A) The wheel s spnnng counte-clockwse nd slowng down. (B) The wheel s spnnng counte-clockwse nd speedng up. (C) The wheel s spnnng clockwse nd slowng down. (D) The wheel s spnnng clockwse nd speedng up? Physcs 07: ectue 6, Pg 7 Physcs 07: ectue 6, Pg 9 Exple: Wheel And ope A wheel wth dus 0.4 ottes feely bout fxed xle. Thee s ope wound ound the wheel. Sttng fo est t t 0, the ope s pulled such tht t hs constnt cceleton 4/s. How ny evolutons hs the wheel de fte 0 seconds? (One evoluton π dns) Exple: Wheel And ope A wheel wth dus 0.4 ottes feely bout fxed xle. Thee s ope wound ound the wheel. Sttng fo est t t 0, the ope s pulled such tht t hs constnt cceleton 4 /s. How ny evolutons hs the wheel de fte 0 seconds? (One evoluton π dns) evolutons (θ θ 0 ) / π nd α θ θ 0 0 t ½ α t (θ θ 0 ) / π 0 ½ (/) t / π (0.5 x 0 x 00) / 6.8 Physcs 07: ectue 6, Pg 0 Physcs 07: ectue 6, Pg Syste of Ptcles (Dstbuted ss): Untl now, we hve consdeed the behvo of vey sple systes (one o two sses). But el objects hve dstbuted ss! o exple, consde sple ottng dsk nd equl ss plugs t dstnces nd. Syste of Ptcles (Dstbuted ss): An extended sold object (lke dsk) cn be thought of s collecton of pts. The oton of ech lttle pt depends on whee t s n the object! K ½ v ½ ( ) Cope the veloctes nd knetc eneges t these two ponts. The otton xs ttes too! K ½ (v) ½ ( ) Physcs 07: ectue 6, Pg Physcs 07: ectue 6, Pg 3 Pge

3 Physcs 07 ectue Syste of Ptcles: Cente of ss f n object s not held then t ottes bout the cente of ss. Cente of ss: Whee the syste s blnced! Buldng oble s n execse n fndng centes of ss. oble Physcs 07: ectue 6, Pg 4 Syste of Ptcles: Cente of ss How do we descbe the poston of syste de up of ny pts? Defne the Cente of ss (vege poston): o collecton of N ndvdul pontlke ptcles whose sses nd postons we know: N y x (n ths cse, N ) Physcs 07: ectue 6, Pg 5 Sple clculton: Consde the followng ss dstbuton: N Y ( x 0 x x 0 )/4 etes (,) X etes X î Y X ( x 0 x x 4 )/4 etes Y 6 etes ĵ Z kˆ (,6) Syste of Ptcles: Cente of ss o contnuous sold, convet sus to n ntegl. y x d d d d whee d s n nfntesl ss eleent. (0,0) (4,0) Physcs 07: ectue 6, Pg 6 Physcs 07: ectue 6, Pg 7 ottonl Dyncs: Wht kes t spn? A foce ppled t dstnce fo the otton xs τ TOT Tng sn φ Toque s the ottonl equvlent of foce Toque hs unts of kg /s (kg /s ) N Tngentl ndl ectue 6, Execse Toque n whch of the cses shown below s the toque povded by the ppled foce bout the otton xs bggest? n both cses the gntude nd decton of the ppled foce s the se. eebe toque eques, nd sn θ o the tngentl foce coponent tes pependcul dstnce A constnt toque gves constnt ngul cceleton ff the ss dstbuton nd the xs of otton en constnt. A. Cse B. Cse C. Se xs cse cse Physcs 07: ectue 6, Pg 8 Physcs 07: ectue 6, Pg 9 Pge 3

4 Physcs 07 ectue ectue 6, Execse Toque n whch of the cses shown below s the toque povded by the ppled foce bout the otton xs bggest? n both cses the gntude nd decton of the ppled foce s the se. eebe toque eques, nd sn φ o the tngentl foce coponent tes pependcul dstnce (A) cse (B) cse (C) se xs cse cse Physcs 07: ectue 6, Pg 0 ottonl Dyncs: Wht kes t spn? A foce ppled t dstnce fo the otton xs τ TOT Tng sn φ Toque s the ottonl equvlent of foce Toque hs unts of kg /s (kg /s) N τ TOT Tng α α o evey lttle pt of the wheel Tngentl ndl Physcs 07: ectue 6, Pg τ TOT α nd net The futhe ss s wy fo ths xs the gete the net (esstnce) to otton Ths s the ottonl veson of TOT Tngentl ndl N Clcultng oent of net whee s the dstnce fo the ss to the xs of otton. Exple: Clculte the oent of net of fou pont sses () on the cones of sque whose sdes hve length, bout pependcul xs though the cente of the sque: oent of net,, (hee s just pont on the wheel) s the ottonl equvlent of ss. f s bg, oe toque s equed to cheve gven ngul cceleton. Physcs 07: ectue 6, Pg Physcs 07: ectue 6, Pg 3 Clcultng oent of net... o sngle object, depends on the otton xs! Exple: 4 4 ( / / ) ectue 6, Hoe Execse oent of net A tngul shpe s de fo dentcl blls nd dentcl gd, ssless ods s shown. The oent of net bout the, b, nd c xes s, b, nd c espectvely. Whch of the followng s coect: (A) > b > c (B) (C) > c > b b > > c b c Physcs 07: ectue 6, Pg 4 Physcs 07: ectue 6, Pg 5 Pge 4

5 Physcs 07 ectue ectue 6, Hoe Execse oent of net () b 3 c () Whch of the followng s coect: (A) > b > c (B) (C) > c > b b > > c b c Clcultng oent of net... o dscete collecton of pont sses we found: N o contnuous sold object we hve to dd up the contbuton fo evey nfntesl ss eleent d. An ntegl s equed to fnd : d d Physcs 07: ectue 6, Pg 6 Physcs 07: ectue 6, Pg 7 oents of net Soe exples of fo sold objects: oents of net... Soe exples of fo sold objects: d Sold dsk o cylnde of ss nd dus, bout pependcul xs though ts cente. Sold sphee of ss nd dus, bout n xs though ts cente. /5 ½ Physcs 07: ectue 6, Pg 8 Thn sphecl shell of ss nd dus, bout n xs though ts cente. Use the tble See Tble 3.3, oents of net Physcs 07: ectue 6, Pg 9 oents of net Soe exples of fo sold objects: Thn hoop (o cylnde) of ss nd dus, bout n xs though t cente, pependcul to the plne of the hoop s just otton & Knetc Enegy Consde the sple ottng syste shown below. (Assue the sses e ttched to the otton xs by ssless gd ods). The knetc enegy of ths syste wll be the su of the knetc enegy of ech pece: 4 K v K ½ v ½ v ½ 3 v 3 ½ 4 v 4 4 Thn hoop of ss nd dus, bout n xs though dete Physcs 07: ectue 6, Pg 30 Physcs 07: ectue 6, Pg 3 Pge 5

6 Physcs 07 ectue otton & Knetc Enegy Notce tht v, v, v 3 3, v 4 4 So we cn ewte the suton: K ] v [ We ecognze the quntty, oent of net o, nd wte: 4 4 K 3 3 Physcs 07: ectue 6, Pg 3 ectue 6, Execse ottonl Knetc Enegy We hve two blls of the se ss. Bll s ttched to 0. long ope. t spns ound t evolutons pe second. Bll s on 0. long ope. t spns ound t evolutons pe second. K Wht s the to of the knetc enegy of Bll to tht of Bll? A. ¼ B. ½ C. D. E. 4 Bll Bll Physcs 07: ectue 6, Pg 33 ectue 6, Execse ottonl Knetc Enegy K /K ½ / ½ 0. / 0. 4 otton & Knetc Enegy... The knetc enegy of ottng syste looks sl to tht of pont ptcle: Wht s the to of the knetc enegy of Bll to tht of Bll? (A) /4 (B) / (C) (D) (E) 4 Bll Bll Pont Ptcle K v v s lne velocty s the ss. ottng Syste K s ngul velocty s the oent of net bout the otton xs. Physcs 07: ectue 6, Pg 34 Physcs 07: ectue 6, Pg 35 So oent of net nd ottonl Enegy K whee Notce tht the oent of net depends on the dstbuton of ss n the syste. The futhe the ss s fo the otton xs, the bgge the oent of net. o gven object, the oent of net depends on whee we choose the otton xs (unlke the cente of ss). n ottonl dyncs, the oent of net ppes n the se wy tht ss does n lne dyncs! Wok (n ottonl oton) Consde the wok done by foce ctng on n object constned to ove ound fxed xs. o n nfntesl ngul dsplceent dθ :whee d dθ dw Tngentl d dw ( Tngentl ) dθ xs of otton dθ dw τ dθ (nd wth constnt toque) We cn ntegte ths to fnd: W τ θ (τ f τ ) θ Anlogue of W W wll be negtve f τ nd θ hve opposte sgn! φ d dθ Physcs 07: ectue 6, Pg 36 Physcs 07: ectue 6, Pg 37 Pge 6

7 Physcs 07 ectue Wok & Knetc Enegy: ecll the Wok Knetc-Enegy Theoe: K W NET Ths s tue n genel, nd hence pples to ottonl oton s well s lne oton. So fo n object tht ottes bout fxed xs: K ( f ) WNET Physcs 07: ectue 6, Pg 38 ectue 6, Hoe execse Wok & Enegy Stngs e wpped ound the ccufeence of two sold dsks nd pulled wth dentcl foces fo the se lne dstnce. Dsk hs bgge dus, but both e dentcl tel (.e. the densty ρ /V s the se). Both dsks otte feely ound xes though the centes, nd stt t est. Whch dsk hs the bggest ngul velocty fte the pull? W τ θ d ½ (A) Dsk (B) Dsk (C) Se stt fnsh d Physcs 07: ectue 6, Pg 39 ectue 6, Hoe execse Wok & Enegy Stngs e wpped ound the ccufeence of two sold dsks nd pulled wth dentcl foces fo the se lne dstnce. Dsk hs bgge dus, but both e dentcl tel (.e. the densty ρ /V s the se). Both dsks otte feely ound xes though the centes, nd stt t est. Whch dsk hs the bggest ngul velocty fte the pull? W d ½ ½ ( / ) ½ nd < (A) Dsk (B) Dsk (C) Se stt fnsh d Physcs 07: ectue 6, Pg 40 Exple: ottng od A unfo od of length 0.5 nd ss kg s fee to otte on fctonless pn pssng though one end s n the gue. The od s elesed fo est n the hozontl poston. Wht s (A) ts ngul speed when t eches the lowest pont? (B) ts ntl ngul cceleton? (C) ntl lne cceleton of ts fee end? Physcs 07: ectue 6, Pg 4 Exple: ottng od A unfo od of length 0.5 nd ss kg s fee to otte on fctonless hnge pssng though one end s shown. The od s elesed fo est n the hozontl poston. Wht s (B) ts ntl ngul cceleton?. o foces you need to locte the Cente of ss s t / ( hlfwy ) nd put n the oce on BD. The hnge chnges eveythng! Σ 0 occus only t the hnge g but τ z α z sn 90 t the cente of ss nd ( (/) ) α z (/) g Exple: ottng od A unfo od of length 0.5 nd ss kg s fee to otte on fctonless hnge pssng though one end s shown. The od s elesed fo est n the hozontl poston. Wht s (C) ntl lne cceleton of ts fee end?. o foces you need to locte the Cente of ss s t / ( hlfwy ) nd put n the oce on BD. The hnge chnges eveythng! α g nd solve fo α z Physcs 07: ectue 6, Pg 4 Physcs 07: ectue 6, Pg 43 Pge 7

8 Physcs 07 ectue / Exple: ottng od A unfo od of length 0.5 nd ss kg s fee to otte on fctonless hnge pssng though one end s shown. The od s elesed fo est n the hozontl poston. Wht s (A) ts ngul speed when t eches the lowest pont?. o foces you need to locte the Cente of ss s t / ( hlfwy ) nd use the Wok-Enegy Theoe. The hnge chnges eveythng! g g W gh ½ W g/ ½ ( (/) ) nd solve fo Connecton wth oton f n object of ss s ovng lnely t velocty V wthout ottng then ts knetc enegy s f n object of oent of net s ottng n plce bout ts cente of ss t ngul velocty then ts knetc enegy s K K Wht f the object s both ovng lnely nd ottng? K V T V Physcs 07: ectue 6, Pg 44 Physcs 07: ectue 6, Pg 45 Connecton wth oton... ollng oton So fo sold object whch ottes bout ts cente of ss nd whose s ovng: Agn consde cylnde ollng t constnt speed. V K TOT V V V Physcs 07: ectue 6, Pg 46 Physcs 07: ectue 6, Pg 54 Exple : ollng oton A cylnde s bout to oll down n nclned plne. Wht s ts speed t the botto of the plne? Bll hs dus Exple : ollng oton A cylnde s bout to oll down n nclned plne. Wht s ts speed t the botto of the plne? Use Wok-Enegy theoe Bll hs dus h θ v? h gh ½ v ½ θ v? gh ½ v ½ (½ )(v/) ¾ v v (gh/3) ½ Physcs 07: ectue 6, Pg 55 Physcs 07: ectue 6, Pg 56 Pge 8

9 Physcs 07 ectue ollng oton Now consde cylnde ollng t constnt speed. V oton Agn consde cylnde ollng t constnt speed. otton only V Tng Both wth V Tng V V V Sldng only V The cylnde s ottng bout nd ts s ovng t constnt speed (V ). Thus ts totl knetc enegy s gven by : K TOT V Physcs 07: ectue 6, Pg 57 Physcs 07: ectue 6, Pg 58 Angul oentu: We hve shown tht fo syste of ptcles, oentu s conseved f p v dp 0 dt Exple: Two Dsks A dsk of ss nd dus ottes ound the z xs wth ngul velocty 0. A second dentcl dsk, ntlly not ottng, s dopped on top of the fst. Thee s fcton between the dsks, nd eventully they otte togethe wth ngul velocty. Wht s the ottonl equvlent of ths? z z ngul oentu s conseved f τ d dt 0 0 Physcs 07: ectue 6, Pg 59 Physcs 07: ectue 6, Pg 60 Exple: Two Dsks A dsk of ss nd dus ottes ound the z xs wth ntl ngul velocty 0. A second dentcl dsk, t est, s dopped on top of the fst. Thee s fcton between the dsks, nd eventully they otte togethe wth ngul velocty. No Extenl Toque so z s constnt f f ½ 0 ½ f z z ectue 6, Oct. 9 Assgnent: Wednesdy s n ex evew sesson, Ex wll be held n oos B0 & B30 n Vn Vleck t 7:5 P P Hoewok 7, Ch., 5 pobles, NOTE: Due Wednesdy t 4 P P Hoewok 7A, Ch. 3, 5 pobles, vlble soon 0 Physcs 07: ectue 6, Pg 6 Physcs 07: ectue 6, Pg 6 Pge 9

10 Physcs 07 ectue An exple: Neuton St otton Neuton st wth ss of.5 sol sses hs dete of ~ k. Ou sun ottes bout once evey 37 dys f / / f / f (7x0 5 k) /( k) 4 x 0 9 gves llsecond peods! Angul oentu s undentl Quntty The concept of ngul oentu s lso vld on subcoscopc scle Angul oentu hs been used n the developent of oden theoes of toc, olecul nd nucle physcs n these systes, the ngul oentu hs been found to be fundentl quntty undentl hee ens tht t s n ntnsc popety of these objects peod of puls s s Physcs 07: ectue 6, Pg 77 Physcs 07: ectue 6, Pg 78 undentl Angul oentu Angul oentu hs dscete vlues These dscete vlues e ultples of fundentl unt of ngul oentu The fundentl unt of ngul oentu s h-b ntnsc Angul oentu Whee h s clled Plnck s constnt h h π kg s 34 nh ( n,,3,...) Physcs 07: ectue 6, Pg 79 photon Physcs 07: ectue 6, Pg 80 Angul oentu of olecule Consde the olecule s gd oto, wth the two tos septed by fxed dstnce The otton occus bout the cente of ss n the plne of the pge wth speed of Angul oentu of olecule (t hets the wte n cowve ove) h h / E h /(8π ) [ J (J) ] J 0,,,. Physcs 07: ectue 6, Pg 8 Physcs 07: ectue 6, Pg 8 Pge 0